Simplify the following expression and state the condition under which the simplification is valid. $x = \dfrac{-9p^2 - 81p - 180}{-2p^3 + 2p^2 + 40p}$
First factor out the greatest common factors in the numerator and in the denominator. $ x = \dfrac {-9(p^2 + 9p + 20)} {-2p(p^2 - p - 20)} $ $ x = \dfrac{9}{2p} \cdot \dfrac{p^2 + 9p + 20}{p^2 - p - 20} $ Next factor the numerator and denominator. $ x = \dfrac{9}{2p} \cdot \dfrac{(p + 4)(p + 5)}{(p + 4)(p - 5)}$ Assuming $p \neq -4$ , we can cancel the $p + 4$ $ x = \dfrac{9}{2p} \cdot \dfrac{p + 5}{p - 5}$ Therefore: $ x = \dfrac{ 9(p + 5)}{ 2p(p - 5)}$, $p \neq -4$